DOE/ET/51013-101
PFC/RR-83-31
A COMPARISON BETWEEN THE RADIAL DENSITY BUILDUP
IN THE TARA PLUGS USING HYDROGEN VERSUS
DEUTERIUM NEUTRAL BEAMS
Donald T. Blackfield
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
November 1983
This work was supported by the U. S. Department of Energy Contract
No. DE-AC02-78ET51013. Reproduction, translation, publication, use
and disposal, in whole or in part by or for the United States government is permitted.
-2-
ABSTRACT
The WOLF code is used to compare the beam divergences from a TARA source
using hydrogen and deuterium.
Factors which influence the divergence which
are investigated are the electron temperature,
positions and ion beam current density.
initial
ion energy,
electrode
The beam divergence for 20 keV
hydrogen is found to be only 20% smaller than for 25 keV deuterium for the
same electrode positions.
found to be independent
Since the optimal positioning of the electrodes is
of mesh spacing,
a large
parameter
study
is
undertaken using little computer time.
A time-dependent radial Fokker-Planck code is next used to examine
radial density buildup in a plug of the TARA tandem mirror.
hydrogen and deuterium neutral beams,
current and energy,
are studied.
For both
the influences of beam positioning,
edge neutral pressure and assumed electron temperature
In TARA,
narrower profile.
the
hydrogen beams produce a higher density with a
In both hydrogen and deuterium,
shifting the beam 3 cm
above the midplane produces the "optimal" density profile.
achieved for edge pressures less than 1 x 10 ->
Torr.
Buildup is
Finally, higher
electron temperatures can cause a lower density buildup due to greater
electron endlosses,
decrease in the electron impact ionization cross section
and increase in the ion loss cone.
-3-
INTRODUCTION
The Livermore tandem mirror experiment TMX has obtained a sloshing ion
distribution when deuterium neutral beams were injected into the plug region
after being unsuccessful with hydrogen neutral beams. This has motivated the
design group of TARA to consider changing from a hydrogen experiment to
deuterium. For the neutral beam group this would mean changing from 20 keV
hydrogen to injecting 25 keV deuterium into the plugs. Several physics issues
must be addressed
in order to consider the overall effect on plasma
performance if this change is made. Several fundamental plasma parameters are
functions of ion mass and velocity.
In TABLE 1 some of these parameters are
shown for 20 keV hydrogen and deuterium and 25 keV deuterium slowing down on
a background plasma with ion energy of 10 keV and electron energy of 150 eV
with a density of 2
mass ratio is
12
-3
. For example with deuterium the charge to
larger. Using Spitzer's formula for the 90 degree scattering
x
10
cm
time [11,
1/2
-
904
m
8
x
3/2
(3kT)
ln(1
0.714wne lnQ
confinement should be better. However the slowing down time for 25 keV
deuterons on a deuterium plasma is also longer [1-2],
-
(2)
1/2
3w3
2(1 + m/mf)ADlf
with
4
Bye nfln
AD
and
m2
P
-4-
m
f
2
i 2 = f(4)
f
mkTf
so less neutral beam energy may be deposited in the plugs.
higher mass and energy results in an increase
The deuterium
in electron drag while pitch
angle scattering is decreased. The Larmor radius for deuterium is larger than
for hydrogen so less Larmor radii "fit"
within the plasma.
Finite Larmor
Radius (FLR) effects are increased and radial profiles become more important.
Several physics issues may be raised when deuterium neutral beams are
used.
Is the amount of cold streaming gas affected? Should one build up to a
higher or lower density with deuterium? Should the larger Larmor radius cause
a shift in the injection angle to obtain the desired density? What is the
effect on the plasma halo with
deuterium? Since the ion gyrofrequency
decreases for deuterium either the rf generating frequency needs to be
decreased or higher harmonic heating will occur.
If the magnetic field is
doubled to heat at the fundamental resonance, how is the plasma affected?
The above are all important considerations which need to be examined.
Since changing from hydrogen to deuterium may have a large impact on the TARA
design we have decided to examine this problem in four steps.
The first
step
is to examine the effect of changing from 20 keV hydrogen to 25 keV deuterium
on the neutral beam source design. The results which will be summarized below
were obtained from the WOLF code [3-4]. It appears that the source does not
need to be redesigned although the beam divergence increases slightly.
The next
step is
to examine the effect of the higher energy deuterium
neutral beams on the buildup of the confining plugs in TARA. This is examined
using a time-depenedent Fokker-Planck buildup code which uses a square-well
approximation for the spatial variation of the ambipolar potential and
magnetic field [5].
This code uses finite
width neutral beams with three
energy components to examine the time evolution of the plasma density and
temperature profiles but assumes uniformity in the axial dimension . The
preliminary results
show that the plasma does not build up to as high a
central density with 25 keV deuterium beams although the plasma is broader.
-5-
However it
is shown that TARA should have enough neutral beam current to
build to a reasonable density.
Next,
a multi-cell,
multi-species,
time-dependent
0-D particle and
energy balance code is being developed to examine the TARA design [6].
This
code will be used to examine the impact of changing from hydrogen to
deuterium using existing theories.
Finally the effects of deuterium on the
various proposed ICRF schemes will be examined using both an existing slab
model code
[7] and a O-D time-
dependent
Fokker-Planck code [8-9].
In
addition a bounce-averaged Fokker-Planck code [10] has been developed that,
with modifcations, can be used to further understand the effects of ICRF in
TARA.
The above questions and methods involve a long-range study-. In this
paper both WOLF and buildup results will be presented.
WOLF RESULTS
The WOLF Code,
a set of programs which calculates ion trajectories in
two dimensions through a set of electrodes,
attempts to minimize the beam
divergence and obtain a self-consistent shape and position of the emitting
surface. Factors which influence the beam divergence which we investigate are
the electron temperature,
ion beam current density.
emitter position,
initial
ion energy,
electrode positions,
and the
For a given ion energy and electron temperature,
ion current density and electrode positions we try to
minimize the beam divergence and deform the emitting surface to achieve a
predetermined electric field at the emitting surface.
FIGURE 1 shows the electrode positions,
trajectories
equipotential
for a 20 keV hydrogen beam calulation.
the problem only half the beams are shown.
lines and ion
Due to the symmetery of
WOLF attempts to deform the
emitter surface to obtain an electric field value near 300 V/cm, consistent
with the calculations by Self [11].
The greater the electric
field
at the
emitter, the smaller the beam divergence. WOLF attempts to obtain a value of
300 V/cm for each mesh interval.
divergence is 0.0236
For this "optimized" case the rms beam
radians and the rms deviation in the electric fields
from 300 V/cm is 291.6 V/cm.
-6-
FIGURE 2 shows the geometry of the emitting surface for the 20 keV
The points marked by the squares
hydrogen and the 25 keV deuterium cases.
show the emitter position for both these cases.
The diamond points show the
emitting surface for a nearly optimized 25 keV deuterium case.
Later we will
examine the sensitivity of our results to the assumed emitting surface.
For the same geometry as
divergence
60.1 V/cm.
the 20 keV hydrogen
case,
the rms
beam
is 0.0277 radians with an electric field rms deviation of only
FIGURE 3 shows the geometry and ion trajectories for this 25 keV
deuterium case.
FIGURE 4 shows the sensitivity of beam divergence
electron temperature.
Again the diamond shaped and the square shaped points
correspond with the two surfaces in FIGURE 2.
increases,
to the assumed
As the electron temperature
the electric field at the surface of the emitter increases while
the beam divergence decreases.
varies with initial
FIGURE 5 shows how the rms beam divergence
ion temperature.
The ions initially are assumed to be
maxwellian, drifting with the electron temperature [3].
FIGURES 6 and 7 show the sensitivity of beam divergence and rms electric
field deviation with beam current.
If only FIGURE 6 was examined one might
draw the conclusion that a lower beam current than the highest achievable
value of 0.390 A/cm2 [12] might be better.
However,
FIGURE 7 shows that the
calculated electric fields are larger than the desired value of 300 V/cm when
the current density decreases.
To obtain a value of 300 V/cm the plasma
surface must be moved away from the extractor electrode and the beam
divergence would then increase.
current density,
Therefore,
if
On the other hand, for the larger values of
the calculated electric
field values are too small.
the emitter surface is moved closer to the extractor to
increase the electric field, the beam divergence would again increase.
FIGURE 8 shows how the beam divergence varies with distance between the
emitter surface and the acel electrode.
At points A, B,
and C the emitting
surface had to be moved so that the code could converge to a solution.
In
general for a given optimized solution the beam divergence will quickly
-7-
increase as the electrode positions are moved relative to the emitting
surface.
The distance between the acel and decel electrodes has a much
smaller effect on the beam divergence.
Finally we have investigated the sensitivity of our results to the
"coarseness"
of the mesh,
hence the number of individual beamlets which we
use in calculating the beam divergence.
FIGURE 9 shows a typical triangular
grid used in the coarse mesh calculations which have been presented. A
typical "fine" mesh grid is also shown in FIGURE 9. A fine mesh 20 keV
hydrogen ion trajectory and equipotential line plot is shown in FIGURE 10.
Five times more beamlets are used to obtain a more accurate solution.
A
comparison of 20 keV hydrogen beam divergences for various current densities
can be seen in FIGURE 11.
finer mesh,
Although the beam divergence decreases with the
the optimal current density remains the same.
Therefore a coarse
mesh, which requires a smaller amount of computer usage,
can be used in a
parameter study to obtain an optimal design.
In conclusion we have found that when 25. keV deuterium beams are used
instead of 20 keV hydrogen beams,
to be redesigned,
divergence
the neutral beam ion source does not need
assuming an increase,
is acceptable.
perhaps as large as 20%,
in beam
The maximum extractable beam current should be
similar since the space-charge current limitation [131,
3/4
1/2
1
2e
9wmad
v
is the same for both cases. Assuming that 25keV deuterium beams can be
successfully injected into the plugs of TARA we next examine the timedependent density buildup of the plugs with these beams.
-8-
BUILDUP RESULTS
A time-dependent,
by Futch and others
multi-species,
Fokker-Planck code has been developed
[5] to calculate f(r,v,t)
using an orbit-averaged
treatment of Coulomb collisions and includes the finite Larmor radius of the
ions.
This model calculates the ion distribution function in the plug region
in one dimension in velocity space (v),
and in the radial dimension in real
space (assuming cylindrical geometery). A perpendicular injected finite width
neutral beam is assumed to be incident a distance d above or below the
magnetic axis. The beam is assumed to be gaussian-shaped in the radial
direction and expotentially decaying axially. Since the neutral beams in TARA
will be injected at 30-40 degrees it
is difficult to determine the actual
beam requirements from only a radial treatment.
the equivalent
Defining cbl and cb2
to be
distances above and below the magnetic axis for a uniform
neutral beam injected perpendicular (0 - 90*)
and assuming that the beam is
gaussian in the z-direction
J(z)
-
2 2
-z /13
(6)
J0 e
the total beam current would be
3
I total
ioti
(7).
i [cbl + cb2]Lz
J
13
= 1.602 x 10
[1.372 x 10
C
e
][1 + 2/3 + 1][10
2
2
dz
-13
- 113A
where cbl=1.5Xcbml and cb2-1.5Xcbm2
and cbml and cbm2 are the 1/e points
above and below the magnetic axis for the neutral beam footprint.
since the beam is injected at 40 degrees,
However,
the axial extent is increased so
the effective beam current needed is decreased by sin(O)
-9-
) - 113
Itotal - 113 x sin(e
x
sin(40 )
(8)
73A
Since the beam particles once they are. ionized bounce along z between the
mirror points, the effective beam density decreases so a larger beam current
is required. In addition the actual gaussian shape of the beam in the zdirection may be larger than EQUATION 6 shows. If we include a factor to
account for these effects we may write EQUATION 8 as
C(r profile, z profile, z bounce pt.) x 113 x sin(O
Itotal-
-
)
(9)
100A
We have decided to pick C so that the total current is 100 Amps.
code only uses J1 , J 2 ,
The buildup
3 so that even though the absolute value of the beam
current cannot be calculated the scaling with beam current can be (for the
same injection angle) determined.
In addition to neutral beams,
the model also takes into account the
presence of Franck-Condon neutrals at the plasma edge. Assuming that these
neutrals comprise 10% of the total cold gas at the plasma edge [14] the edge
pressure is given by
P M fcnum
3.2 x 105
(10)
Torr
where fcnum is an input.
This model has been modified to take into account the time-dependent influx
of cold gas due to the neutral beam injection. The following expression
approximates the assumed neutral gas rise at the plasma edge [6]
- fcnum(1
n
fc
-
-4t
+ .001)
e
t
<
5ms
(11)
-10-
and is plotted in FIGURE 12.
The initial plasma parameters are given in TABLE 2. We first compare the
buildup of a TARA plug cell using the two base cases; 20 keV hydrogen beams
and 25 keV deuterium beams. Both cases assume 100 Amps 'on target and that the
10
-8
[10] equals 3.2
10
).
In all cases anomalous electron heating (ECRH) or energy transport was
edge pressure initially is 10
Torr (fcnum in Eq.
x
assumed to keep the electron energy fixed. For most cases, E e100 eV,
although we also examined E - 250eV, 500eV, 750eV and 1000eV. Fixing Ee often
resulted in the code stopping when the ambipolar potential requirements could
In the other cases,
not be satisfied after a set number of iterations (200).
With a time varying
the simulations were for 400 time steps when t-10.0 ms.
edge pressure, no cases reached equilibrium. However by 4 ms (200 time steps)
our base cases had sufficient density buildup and run longer than the global
cofinement
time of 2 ms.
To determine how the density buildup varies with
species, neutral pressure, E , and beam parameters, we will examine all cases
e
at the same point in time,
t- 4ms (200 time steps) which is earlier than the
occurrence of any ambipolar potential problems and is larger than the global
confinement time.
FIGURES 13-14 show that the initial ion density and energy profiles are
cases,
the same number of velocity and spatial
slightly different.
In all
mesh points are used.
Since deuterium has a greater mass,
and spatial points are further apart.
both the velocity
This introduces a slighlty larger
numerical inaccuracy when calculating densities and energies at a given
-3
12
radial point. For hydrogen, the peak density is 1.8 x 10
average density of 6.0
x
10
10
cm
with an
-3
cm
while the peak deuterium density is
-3
11
-3
12
1.4
11
x
cm
with average density of 4.4 x 10
. In both cases this
cm
is lower than the inputed starting peak value of 2.0
x
10
12
cm
-3
. We have
assumed that the initial distribution is gaussian shaped. However an ion
distribution in the plug cannot be gaussian in the presence of a loss cone.
The code assumes that all particles in the loss cone are instantaneously lost
from the system. Therefore the overall density is decreased because particles
-11-
are "deposited" into the loss cone.
dependence
In addition, there is an initial radial
2
n(l)a(1 - r./r
2
(v))
(12)
r (v) is the radial point where r + P,i(v) = r = 14.75 cm. Therefore r
pp
pi
pi
pi
is the plasma radius and is a function of velocity. Any particle whose
perpendicular velocity results in a Larmor radius excursion beyond the
maximum radial grid point is lost thereby decreasing the actual plasma radius
for this velocity at this radial point. Since deuterons have a larger larmor
radius than protons, their initial density is smaller.
FIGURES 15-16 show the time evolution of the ion density for the two
base cases (see TABLE 3). Although the hydrogen case starts with a slighlty
12
-3
higher initial density, at 4 ms, the average density is 5.4 x 10
cm
while
the deuterium density is only 2.2 x 10 12 cm -3. However, the larger deuterium
Larmor radius causes the density profile to be broader; the radius at which
the electron density equals 1.0 x 10-4 times the peak density is 14.7 cm for
deuterium and is 14.0 cm for hydrogen. The hydrogen global particle
confinement time is also larger, 2.2 ms as compared to 1.7 ms for deuterium.
The dominant system power losses in both cases are electron and ion end
losses and charge exchange from the Franck-Condon neutrals. For hydrogen,
these values are 9.0 kW, 23.5 kW and 4.8 kW while for deuterium, they are
7.4 kW, 2.1 kW and 3.0 kW. The electron endloss is the pastukov loss and is
defined as [131
-1
2
ne 2R
e
T )T
=1/2 r 2R+1 ln(4R+2)
I
5
+
T
-.
1 T
1
()
The ion endloss is determined from the loss of particles to the loss
cone. In all cases the ions are assumed to initially
have a uniform
temperature profile with an energy of 10 keV. An energy source is the
ionization of the neutral beam by ion and electron impact and the charge
-12exchange of neutral beam atoms with energies
larger than 10 keV.
Charge
exchange with atoms below 10 keV causes a loss of energy. The trapping power
is the sum of the charge exchange and ion and electron ionizations.
For the
two base cases the total trapping power for hydrogen is 143 kW with 13 kW of
net power coming from charge exchange.
For the deuterium case only 32 kW of
beam power is trapped with 2.2 kW coming from charge exchange.
increase
in trapping power is due to the higher initial
hydrogen case.
Some of the
density in the
In the APPENDIX the cross sections and av's
exchange and ion and electron ionizations are shown.
for
charge
Although the av for the
trapping of 25 keV deuterium neutral beams is nearly the same as for 20 keV
hydrogen the lower energy components of the 20 keV hydrogen beam have a
higher trapping av than the half and third energy components
of deuterium.
Consequently more of the hydrogen neutral beam is trapped which leads to a
higher density which in turns leads to a higher trapping rate.
energies,
At these low
charge exchange is the largest contributor to the trapping cross
section. FIGURE A-1 in the APPENDIX shows that the hydrogen charge exchange
cross section is larger. The ion impact ionization for hydrogen is smaller.
The electron impact ionization cross section is also small and is the same
for both species.
This
cross section depends only upon the electron
temperature since the electrons are assumed to travel much faster than the
neutral beam atoms.
Finally,
the vessel walls are considered far enough
removed from the plasma so that wall losses (~1
W) can be ignored.
FIGURES 15-16 also show the time evolution of the ion energy profiles.
In both cases ion heating (ICRF) was assumed to be present so that the
initial
ion energy was 10 keV (T -6.7
at 4.0 ms.
keV,
In the hydrogen case,
keV) which is close to the ion energy
the peak energy at the edge is
which is higher than the neutral beam energy.
the ion density is less than 6.0
8
x
10
However,
almost
21
at this point,
-3
cm
at r=13.75 cm. This somewhat high
value may not be physical (up scattering or interaction with the radial
potential profile) but could be numerically produced.
may still
continue
to rise,
Although the density
FIGURES
15-16 show that the ion temperature
profile quickly reaches a steady-state.
The electron energy is held constant
at 100 eV so there is a large flow of energy from the ions to the electrons.
-13Finally FIGURE 17 shows the time evolution of the log of the density
profile of the Franck-Condon neutrals for both cases.
rises,
As the central density
the hydrogen Franck-Condon neutrals are expelled from the center.
Their density rises at the edge due to the assumed increase in edge pressure
with time. At t=O ms, there is assumed to be no Franck-Condon neutrals in the
plasma.
In the deuterium case,
the central density is not high enough to
significantly expel the Franck-Condon neutrals.
of magnitude
There is less than an order
decrease in neutral density from the edge to the center.
Although there is more neutral gas charge exchange power loss in the hydrogen
gas case (since there is more system energy)
the actual amount of charge
exchange power loss from the center is approximately three times smaller than
the deuterium case.
In the hydrogen case,
with 100 Amps of
average density increases from 6.0
with a peak value near 2.2
increases from 4.0
x
10
11
x
-3
cm
10
13
10
x
cm
to 2.2
x
-3
10
11
20 keV of neutral beams,
cm
-3
to 1.1
x
10
13
-3
cm
while for deuterium, n
e
the
in 4 ms
average
12 cm-3 with a peak near 6.2
x
10 1
-3
. In both cases there is a sufficient buildup of density in 4 ms. The
buildup code is optimistic since it has only radial variations. However, TARA
cm
should have over 150 Amps of beams so there is a rather wide margin to allow
for buildup. Having examined in detail our two base cases,
we now proceed to
examine how the density varies when neutral beam position,
energy,
current,
neutral edge pressure and the fixed electron temperature are varied.
DENSITY BUILDUP AND BEAM POSITION
FIGURES 18-20 show the hydrogen ion density,
energy and Franck-Condon
profiles at t=4.0 ms for several values of d from -3
cm to 10.5 cm. As d
increases, the peak ion density increases. The ion energy remains fairly
constant except for d- 10.5 cm.
As FIGURE 20 shows, the Franck-Condon
neutral density at the center becomes appreciable for d < 1 cm.
values of d,
there is little
density buildup.
versus d is plotted for these cases
For these
The average electron density
in FIGURE 21.
density increases with d until d gets too large.
We see that the average
When a charge exchange or
-14ionization event
occurs,
the finite
Larmor radius of the beam shifts the
particle guiding center towards the minus d-direction. Therefore, by shifting
the beam above the axis, more beam particles are trapped. However, if the
beam is shifted too far above the axis, the radial decrease in density
produces
a decrease in beam trapping (shown in FIGURE 22).
In addition,
the
profile becomes very peaked. Besides peaking the density and generally
increasing n , the plasma radius shrinks from 14.8 cm to 13.0 cm as d
e
increases. The beam charge exchange power increases with d from 0 kW to 50 kW
until d gets too large when it drops to 10 kW. The higher average and peaked
density causes more of the beam to be trapped.
and peaking density has little
The shrinking plasma radius
effect on the density-squared weighted global
confinement time defined as
fn 2d3v
f<t
>
p
v
(14)
f An 3
<n >
d v
e jAt
and
<n > -i
e
ne d
fd 3v
v
which is between 1.8-2.5 is. The increase in density is also followed by
increases in both the ion and electron endloss terms and charge exchange with
the neutral gas, shown in FIGURE 23.
When 25 keV deuterium beams are used, the behavior of the density
buildup with beam position is slightly different. The density profile is
broader and the plasma radius is less sensitive to beam position. The plasma
radius is 14.7 cm which is larger than for hydrogen. The larger Larmor radius
tends to "smear"
out the effect of moving the finite width beam.
As FIGURES
24-25 show, as d increases, the average density and peak density again
increase.
However,
the larger Larmor radius causes a peaking of the density
off axis as d increases.
For d < -3
cm, there is little
density buildup.
FIGURE 26 shows that for d < 1 cm the ion energy is lower than the initial 10
-15-
keV.
For d=10.5 cm,
there is significant beam trapping (60
ion energy is significantly higher near the edge.
kW)
so that the
FIGURE 27 shows that there
is sufficient density for d > 1 cm to prevent a large fraction of FranckCondon neutrals from penetrating to the center.
Except for the
highest buildup case (d-10.5
cm)
electron or the ion endlosses.
there is little
change in either
the
The higher ion temperature and density for
d-10.5 cm results in more electron (18 kw) and ion (4.5 kW) endloss power. As
d increases, the Franck-Condon neutrals in the plasma density decreases and
this results in a smaller fraction of the total energy loss through charge
exchange on the cold gas (shown in FIGURE 28). As d increases from -0.5 cm to
10.5 cm,
the average ne increases 7 times while the charge exchange of the
gas increases
only 4 times.
In addition,
trapping power increasing with density,
confinement time remains
d-10.5 cm,
we again see in FIGURE 29 the
hence d .
The global particle
from 1.6-1.8 ms except for the higher density,
case where the off axis peaking in density results
in a r
=
0.45
p
ms.
Although the beam trapping power increases from 0.5 kW to 6 kW as d
increases from -5 cm to 5 cm, for d-10.5 cm, there is a sharp drop in the
charge exchange power from the beam to 0.4 kW.
Because the ion energy is
close to 20 keV near the edge where the beam is trapped,
the full
energy
component does not add much energy to the plasma while charge exchange with
the half and third energy components causes a decrease in plasma energy. The
dominant power contribution is from ionization of the beam which increases
with both density and ion temperature.
For both hydrogen and deuterium, a neutral beam of width 7 cm displaced
3 cm above the midplane provides a sufficient density buildup within 4 ms and
density peaked on the midplane with a relatively small density gradient.
Consequently, on changing from 20 keV hydrogen to 25 keV deuterium, the
neutral beamline does not need to be repositioned, provided that a slightly
broader (14.75 cm as compared with a 14.2 cm) but lower density (2.2
cm
-3
as compared with a 5.8
x
10
12
cm
-3
) plasma is tolerable.
x
101 2
-16-
BUILDUP VERSUS-NEUTRAL BEAM CURRENT
We next examined the scaling of the TARA plasma buildup with beam
current fixing the hydrogen full energy component at 20 keV and the deuterium
at
25 keV.
The ratio of full to half to third energy component beam
densities remains 1:2/3:1. This corresponds to a beam energy mix at the
source of 60% full
energy,
20% half energy and 20% third energy.
The H 2(D
)
half energy molecules split into two H(D) ions while the third energy H (D )
3
molecules
split
into three H(D)
beam currents of 50A, 75A,
ions.
100A (base
We investigate buildup for incident
case),
125A and 150A. FIGURES 30-31
show that the density buildup is proportional to beam density. However, this
increase is nonlinear since the the beam current varies from 50-150 A while
the average density changes from 1.9
x
10
12
-3
cm-
to 23.1
x
10
12
-3
cm
The nonlinear behavior in the density buildup is also shown in the total
beam trapping power plotted in FIGURE 32. The amount of power to the plasma
from charge exchange of the beam is similar rising from near 0 kW at 50A to
almost half (259 kW out of 566 kW) at 150A. The ion temperature profiles do
not change significantly with beam current while FIGURES 33-34 show that with
the rising density, the amount of neutral gas, hence the amount of power lost
to charge exchange increases at a smaller rate (2.4 kW up to 5.1 kW) than the
increased energy -content in the plasma. The global particle confinement time,
decreases as the density increases (FIGURE 35), since there are more particle
interactions.
However nc
increases with increasing current. As the current
p
increases from 100A to 150A the plasma radius decreases from 14.45 cm to 12.8
cm.
The higher peak density causes
the 1.0
x
10
-4
x
n
point to shift
inwards.
Similar results are obtained for 25 keV deuterium. FIGURES 36-37 show
the density profile and average density for deuterium.
nonlinear increase in average density (from 0.8
10
12
cm.
-3
x
10
12
cm
-3
There is again a
for 50 A to 8.0
x
for 150 A) however the plasma radius again remains constant at 14.7
In all
cases the large Larmor radius creates a broader plasma with a
cm
-17-
slight density peak off the midplane (near the injection point r= 3 cm).
The
ion temperature profile is again unchanged with beam current remaining near
10 keV with a slight dip at the center and peaking up to 15 keV near the
edge. FIGURES 38-39 show a similar behavior for the franck-condon neutral
penetration and the resultant charge exchange power loss. The charge exchange
power loss varies linearly with beam current rising from 1.4 kW at 50 A to
4.9 kW for 150 A of beam current.
FIGURE 40 shows the increase of beam
trapping power with beam current. The fraction of charge exchange power which
trapping power of the beam is lower than f or
contributes to the total
hydrogen,
being from 5-10% as compared to 10-45% for the higher current
hydrogen cases.
charge exchange
The ion ionization av is higher at higher energies while the
av for deuterium tends to trap a higher ratio of low energy
components to full energy ions.
In additon the larger Larmor radius causes a
greater fraction of the full
energy ions to leave the plasma.
The broader
plasma profile also causes a higher percentage of the charge exchange events
Again although the global confinement time
to occur near the plasma edge.
decreases from 3.3 ms to 1.1 ms, nr
doubles when the current changes from 50
p
A to 150 A.
It is obvious from above that increasing the beam current will result in
higher densities and increased nT . The increased in density is greater than
p
a linear variation.
For deuterium which has a broad profile,
the plasma
radius does not change. For hydrogen the rise in central density also results
in a somewhat smaller radius. The next effect we examine
is,
on keeping the
current constant, how the density buildup scales with beam energy.
BUILDUP VERSUS NEUTRAL BEAM ENERGY
In this section we examine how both hydrogen and deuterium density
buildup scales with beam energy. We have kept the total current constant at
100 Amps and have kept the ratio of full to half to third energy components
the same while the full energy component varies from 15 keV to 30 keV. FIGURE
41 shows that the peak density varies inversely with beam energy.
The lower
the beam energy, the more beam trapping occurs. The average density, shown in
FIGURE 42 shows a weak dependence with beam energy until the beam energy
-18-
approaches
30 keV.
At 30 keV,
the- beam energy is so high that much less
trapping occurs and the average density falls. The effect of beam energy on
the ion temperature can be seen in FIGURE 43.
higher the ion energy until E= 30keV.
for small E (15 keV),
The higher the beam energy,
the
The plasma radius is largest (14.7
cm)
the case where the most beam is trapped.
more beam is trapped
In this case
further from the center and the plasma is somewhat
larger. When the beam energy is raised,
the plasma radius shrinks to 14.2 cm
for 30 keV neutral beams. The beam trapping power (P o 140 kW) remains
constant until E- 30 keV,
where there is a sharp drop to 40 kW.
shows that except for E-30 keV there is little
Franck-Condon neutral penetration,
FIGURE 44
difference in the amount of
the amount of energy being lost to charge
exchange with cold neutrals being from 3.6-4.7 kW until E= 30 keV where a
higher percentage of total power (1.7
kW)
is
lost.
The global confinement
time is relatively insensitive to beam energy being between 2.0 ms to 2.4 ms.
Since deuterium has a larger Larmor radius, the variation in beam energy
has a greater influence on the plasma buildup. In addition to the decrease in
the trapping rate as the energy is increased, the density shown in FIGURES
45-46 decreases from an average n
e
beam to 1.8
energy,
10
x
12
-3
cm
of 6.3
x
10
cm
-3
for a 15 keV neutral
for a 30 keV beam. The Larmor radius increasing with
produces a broader plasma (shown in FIGURE 45). The ion temperature
increasing with beam energy is shown in FIGURE 47.
has little
12
The change in beam energy
effect on the power loss to the cold neutrals
(2.8-3.4 kW)
although the higher peak density excludes more of the neutrals in the plasma
core
shown in FIGURE 48.
The beam trapping power plotted in FIGURE 49
decreases with increasing energy since the density decreases.
The power
gained from charge exchange with the deuterium beam falls from 8.7 kW to 1.7
kW as the beam energy varies from 15 keV to 30 keV.
exchange cross section (see APPENDIX)
sections,
Although the charge
is larger than the ionization cross
the net amount of power gained through charge exchange events is
greatly reduced since hot plasma particles are lost in this interaction. The
global confinement
time remains
fairly
drops,increasing from 1.5 ms to 1.75 ms.
confinment, ni
constant
However,
decreases as the beam energy increases.
as
the
density
the actual measure of
-19-
It
can be seen that in both the hydrogen and deuterium cases,
the lower
beam energies produces a higher density but at slightly lower temperatures.
To buildup in a shorter time or to a higher density,
current at the expense of beam energy.
one should increase the
The effect of changing the energy has
a lesser impact in a hydrogen plasma where the Larmor radius is smaller.
However near 30 keV the Larmor radius is large enough to cause a drop in the
buildup density.
There appears
to be little
gained by increasing the beam
energy from 15 keV to 25 keV and this increase could result in a lower
density plasma if
the beam current is significantly reduced. The deuterium
case shows a monotonic decreasing density as the beam energy is increased.
With both species nr
decreases as beam energy increases.
p
BUILDUP VERSUS NEUTRAL GAS PRESSURE
In this
section we examine how the TARA plug density buildup is
influenced by the time-dependent neutral gas pressure at the plasma edge (see
8
EQUATION 11). In this study we have varied fcnum from 3.2
x
10
11
to 2.0
x
10
which corresponds to an edge neutral density (see EQUATION 10) at t-O ms from
1.0 x 10
Torr to 6.0
x
10-8 Torr and rising exponentially to 17 times this
value at t-4 ms.
FIGURE 50 shows that there is little
affect on the hydrogen density
buildup as the neutral gas pressure is raised from 1.0
10-
x
10
-10
Torr to 3.0
x
Torr.-- The central density begins
to decrease when the neutral pressure
8
at the edge is initially 6.0
x
10-8 Torr. The franck-condon neutral profiles
are shown in FIGURE 51. As the neutral pressure increases, the plasma shrinks
from 14.5 cm to 13.7 cm. The beam trapping power remains fairly constant near
145 kW until the higher pressure causes the central density to drop which in
turn results in a decrease
in trapping power (112
kW).
In the highest
pressure case the ion temperature is somewhat cooler at 4 ms (shown in FIGURE
52).
The global confinement time decreases from 2.4 ms to 1.65 ms resulting
in a decrease
in nr
over this range of approximately the same ratio.
In
p
this range the power lost through charge exchange on the cold gas increases
-20from 20 kW to 25 kW. For hydogen, we can conclude that an initial edge
pressure as high as 6.0
x
10-8 Torr, rising to 1.0
x
10
Torr after 4 ms can
be tolerated with only a slight decrease in plasma density. Unfortunately, it
appears that the deuterium plasma, with its broader and lower density profile
is more sensitive to the edge pressure.
FIGURES 52-54 show that both the central and peak density decreases
the edge neutral pressure is increased.
At the highest initial
there is very little buildup, the average n
e
at t=O to 8.0
10
11
as
edge pressure
increasing from 4.8
x
10 11cm-3
-3
at t=4 ms and actually decreases below this value
due to the increasing edge pressure for t > 4 ms. It appears that with our
x
cm
model an edge pressure of 1.0
plasma to buildup.
x
10
In the pressure
-6
range we have examined neither the ion
temperature nor the plasma radius
density as the pressure increases,
decreases
Torr at t=4 ms is too high to allow the
(14.7
cm)
changes.
With the decrease in
the beam trapping power (see FIGURE 55)
from 36 kW to 11 kW while the amount of neutral gas which reaches
the center as well as the power lost from the resultant charge exchange
increases. Unlike hydrogen, the increased penetration of the edge neutrals
causes a significiant decrease in nT
over this pressure
range.
An initial
edge pressure of 6.0 x 10-8 Torr appears to be too high to achieve a density
buildup in TARA. If the edge pressure is found to be difficult to control, a
likely prospect when neutral beams are used,
achieved with hydrogen.
a better buildup should be
Initial edge pressures greater than 10
too high for either species.
Torr may be
In all of the previous cases, we have kept the
electron energy fixed at 100 eV.
We next vary
the
electron energy
to
determine its affect on the density buildup.
BUILDUP VERSUS ELECTRON TEMPERATURE
In this section we examine the buildup of the TARA plug when the
electron energy is varied from 100 eV to 1000 eV. FIGURES 56-57 show that for
hydrogen, both the peak and average density drops as the electron temperature
increases. The ion temperature profile is unchanged but the neutral gas
-21-
penetration
shown in FIGURE 58
temperature.
The APPENDIX shows that the electron impact ionization av
increases with
increasing electron
decreases with increasing electron energy above 200 eV. The total beam
trapping power decreases with the decreasing cross section and density (see
FIGURE 59). In addition, the higher the electron temperature, the greater the
electron endloss power (increasing
from 10 kW to 70 kW).
There
is a small
decrease in the ion endloss power from 26 kW to 8 kW since the total ion
density decreases. The total system energy has also decreased as the electron
temperature
is increased.
Although the power loss through charge exchange
with neutral gas decreases from 4.3 kW to 2.4 kW, the fraction of the total
system energy lost actually increases.
The plasma radius remains relatively
constant as the electron temperature increases while the global confinement
time increases slightly from 2.2 ms to 2.7 ms.
However,
nr decreases as the
p
electron temperature increases. The combination of increased electron endloss
and decreasing electron impact ionization as well as an enlarging of the ion
loss cone produces a lower density plasma.
The average and peak deuterium density profiles for the various electron
temperatures are shown in FIGURES 60-61.
The same qualitative
behavior of
density buildup with fixed electron temperature can be seen. Again the lower
density profiles have a correspondingly large neutral density as shown in
FIGURE 62. Increasing T
e
with only a slight
increases the electron endloss (from 10 kW to 20 kW)
decrease in the ion endloss (from
2.1 kW to 1.6 kW).
Again the beam trapping (FIGURE 63) and beam charge exchange power (from 2.2
kW to p 0 kW) is seen to decrease while there is also a small decrease in the
power lost from charge exchange with the neutral gas (from 2.9 kW to 2.0 kW).
As for the case of hydrogen,
the plasma radius is insensitive to the
fixed electron temperature and although there is a slight increase in the
global confinement time from 1.7 ms to 2.3 ms there is an overall decrease in
with increasing electron temperature. It appears that for buildup in
p
density in the plug, there is no advantage in heating the electrons to
nT
energies above 100 eV.
The ion energy,
these relatively low electron temperatures.
being at 10 keV is not affected by
In addition,
the decrease in the
electron impact ionization and increase in both the electron endloss and
-22-
enlargement of the ion loss cone boundary tends to inhibit the density
buildup.
CONCLUSION
We have first
examined the beam divergence and electric field at the
emitter surface for a 20 keV hydrogen and a 25 keV deuterium neutral beam
source.
We have shown that there does not have to be a repositioning of the
grid system when 25 keV deuterium is used in a source originally designed for
20 keV
hydrogen.
divergence of 20%.
However there will be a slight increase in the beam
We have examined the beam divergence as both the source
electron and ion temperatures are varied as well as the electrode positions.
We have seen that a higher beam current density results in a smaller beam
divergence.
For a given set of electrode positions,
there is an optimal beam
For hydrogen with a current density of .390 A/cm2, the beam
rms beam divergence is .0185 radians which increases to .0240 radians when 25
current density.
keV. deuterium is used.
The electric field near the electrodes
low enough so that breakdown or arcing should not occur.
appear to be
In addition,
the
change in beam divergence with change in species and voltage agrees with the
space-charge current limit (see EQUATION 3). From the neutral beam source
viewpoint,
there is little
change in going from 20 keV hydrogen to 25 keV
deuterium beams.
We next examined the density buildup of a TARA plug using a 2-D in
velocity space,
1-D in real space,
time-dependent radial Fokker-Planck code.
We observed that with the larger finite Larmor radius of 25 keV deuterons,
the plasma is broader but not as dense at the same point in time. We found
that the plasma builds up faster when the beam current is increased while an
increase in beam energy results in less beam trapping with a resultant
decrease in plasma density.
A hydrogen plasma can buildup with a higher
neutral edge pressure since its density profile is narrower and results in a
better screening of neutrals from the center. An edge density at 4 ms in the
range of 10
-6
-5
Torr can be tolerated for a deuterium plasma with 10
Torr
range for a hydrogen plasma. In both cases, a beam displaced 3 cm above the
midplane produces a reasonably peaked plasma profile. It appears that heating
-23-
the electrons above 100 eV does not improve the density buildup and may
actually degrade it
due to enhanced electron end losses and enlargement of
the ion loss cone.
We can conclude that for TARA more current and lower beam energy results
in a higher density, although lower temperature plasma.
-24-
LIST OF REFERENCES
[1
Lyman Spitzer Jr., Physics of Fully Ionized Gases, 2nd Edition, (1962),
p. 133.
[2]
Lyman Spitzer Jr., Physics of Fully Ionized Gases, 2nd Edition, (1962),
p. 135.
[3]
W. S. Cooper, K. Halboch, S. B. Magyary, Proc. of the Symp. on Ion
Sources and Function of Ion Beams, (Berkeley, 1974), LBL 3399 paper
[4]
W. S. Cooper, A. C. Paul, Wolf a Computer Ion Bear Simulation Package
(unpublished) August 1976.
[5]
A. M. Futch, A Computer Model for the Plasmas Confined by Magnetic
Mirror Fields (unpublished).
[6]
J. Kesner, B. McVey, R. Post, D. Smith, Construction of TARA Tandem
Mirror Facility with an Ion Anchor, (Nov. 1980), (Rev. March 1981).
[7]
D. Blackfield and B. Blackwell, 5th Topical Conference on RF Heating,
(Madison, 1983).
[8]
D. Blackfield and J. Scharer, Nuclear Fusion, (Jan. 1982).
[9]
A. A. Mirin, LLL Report UCRL-51615 Rev. 1 (Feb. 1925).
[101
G. D. Kerbel, 5th Topical Conference on RF Heating, (Madison, Feb. 1983).
[11]
S. A. Self, Phys. of Fluids 6 (Dec. 1963), p. 1762-1768.
[12]
J. Coleman, R. Torti, Private Communication.
[13]
J. Conrad, Rev. Sci. Instrum. 51 (4), (Apr. 1980), p. 418-424.
[14]
Kesner and Post, Private Communication.
-25-
APPENDIX
The buildup code uses the following analytic expression for the charge
exchange of atomic hydrogen
0.6937x10
=
-14r2
1i - 0.155 log 10 (E/p) 2
-(A-i)
ex
a
1 + 0.1112x10A1
(E/p)3
where E is the ion energy in eV and P is the ratio of the hydrogen species'
mass to the proton mass.
FIGURE A-1 shows the charge exchange cross section for ion energies from
1 keV to 50 keV for both hydrogen (solid curve) and deuterium (dashed curve).
The ion impact ionization cross section is given by
-
- 0. 8712[log10(E/p)]a - 8.156 log 10 (E/p) + 34.833]
where E < 150 keV.
FIGURE A-2
(A-2)
shows the cross section for both hydrogen
(solid) and deuterium (dahsed).
The total trapping cross section plotted in FIGURE A-3 for hydrogen and
deuterium is comprised of charge exchange and ion and electron
ionizations.
impact
From FIGURE A-3 we see that for hydrogen energies below 30 keV,
the trapping av is higher.
On comparing the trapping of 20 keV hydrogen
beams with 25 keV deuterium beams,
we first
notice that trapping av for 20
keV hydrogen and 25 keV deuterium are nearly equal.
However, the trapping av
for the half and third energy beams components are higher for hydrogen.
the same initial
target density,
For
one would expect a higher density buildup
using a 20 keV hydrogen beam.
FIGURE A-4 shows that the charge exchange av for hydrogen is higher at
the lower energies compared with deuterium.
However,
charge exchange merely
-26-
replaces a plasma ion with a beam ion, the total ion density remaining
constant.
FIGURE A-5 explains the difference in trapping behavior between
hydrogen and deuterium.
the
For all three components of the 20 keV hydrogen beam
ion impact ionization av is larger.
The increase in ion impact
ionization produces the higher density in hydrogen.
Finally we can partially explain the behavior buildup with the assumed
electron temperature by examining the electron impact ionization
av.
The
electrons are assumed to be Maxwellian with a velocity much greater than the
ion velocity.
Consequently, the ion can be assumed to be at rest so there is
no ion mass dependence in the electron impact ionization cross section.
The
av is therefore a function only of the electron energy and is given by
<av>
- ADl - e
-8
where A = 2.9297 x 10
;
E
I
= 0.246862 x 10
A
<av> e
0
+ BE - CE
-10
(A-3)
-14
;C - 5 x 10
for E < 450 eV
[ln E+B]
(A-4)
e
where A - 1.62 x 10
For E
=
100 eV,
-7
and B
-
2.0618 for E > 450 eV.
-8
the electron impact ionization <av> = 3.2 x 10
all ion energies between 1 and 50 keV.
ion impact ionization cyv.
raising T
e
This av is comparable to that of the
Since charge exchange does not change the plasma
density and ion impact ionization
temperature
cm'/s for
is
not a function
of
the electron
decreases the electron impact ionization hence
decreasing the beam trapping.
In addition,
since the electron
impact
ionization is not a function of ion mass, the increase in density with
hydrogen can be partially explained by the higher ion impact ionization cross
section at lower energies.
-27TABLE 1
FORMULA
UNITS
Mass
M
p
MHz
6
3
3
cm/sec
1.95x10 8
1.56x10
cm
3.62
5.73
m
cm/sec
6.16x 10
4.3x10
4.3x10
(T /m)1/2
e
cm/sec
2.4x10
1.7x10
1.7x10
MHz
296
210
210
sec
1
2
1.4
sec
.52
1.0
.73
sec
2
cm
2
cm
.95
2.7
1.9
*
.
= (E/m)1/2
-1/2
8
-1/2
fpi f
*m
r
900
1/2 3/2
T
T3/2
8s - m 1/2
2 3/2
a m T
T
eq
2
20
m()1/2
C
2
25
m-1/2
VA
D
20
fci
Vthi
D
proton
keV
Energy
V
H
8
8
1.39x10
8
5.13
8
-16
8
7
-16
9.84x10
-17
7.lx10
-7
1.36x 10
-9
9.84x10
<v> ion(See Appendix)
cm /sec
8.9x1016
-16
-16
9.1 x 10
1.4x106
-7
-7
1.28x 10
1.39x 10
-8
-8
2.63x10
1.4x10
<av>
3
cm /sec
3.4x10
3.4x108
3.4x108
ratio to
20 keV H
1
.988
.707
radians
.0236
.0277
acx (See Appendix)
(See Appendix)
a
ion
<av>
cx
i
(See Appendix)
ion
(See Appendix)
space charge J a V 3/
m
Neutral Beam
RMS divergence
3
cm /sec
3
6.6x106
-28-
TABLE 2
25 keV Deuterium Neutral Beam Base Case
Mesh points in x-direction
65
Mesh points in y-direction
15
Width of grid
.381 cm
Length of grid
.98 cm
Extractor grid voltage
25 keV
Acel grid voltage
-2 keV
Decel grid voltage
0 keV
Distance from extractor to acel
.64 cm
Distance from acel to decel
.21 cm
Desired electric field at emitter surface
300 V/cm
Position of beam divergence measurement
.9712 cm
Number of beams
15
Number of beamlets per beam
5
Initial ion v
x
Initial ion v
y
Plasma T
e
RMS beam divergence
Current density
(-.361, -.139, 0., .139, .361) x 3.9 x 10
(1.0) x 3.9
x
6
10
cm/sec
cm/sec
15 eV
.0277 radians
.390 A/cm 2
-29-
TABLE 3
Buildup Base Cases
H
D
Bfield (Gauss)
4000
4000
Mirror ratio
2.5
2.5
Initial plasma radius
14.5 cm
14.5 cm
Plug length
26 cm
26 cm
Radial mesh points
30
30
Ion velocity mesh points
40
40
Electron velocity mesh points
101
101
Initial anbipolar potential (eV)
300
300
E (keV)
e
.150
.150
E (keV)
10
10
mass (m/mH)
1
2
Beam injection energies (keV)
20;10;6.7
25;12.5;8.3
1:2/3:1
1:2/3:1
Total "effective" current (Amps)
100
100
Total "effective" power (MW)
1.25
1.56
Initial n (0) (cm- 3)
1.8x10 12
1.4x1012
6.OxlO1
4.4x1011
Beam current density (xl.372x10
Initial n
e
(cm 3 )
18
cm
-2
sec
-1
)
-30TABLE 3 --
CONT'D
At t-4ms
D
H
13
-3
)
12
n (0) (cm
2.1x 103
6.3x10
n
(cm-3)
5.7x1012
2.2x1012
T
(Ms)
2.2
1.7
R
(cm)
14.0
14.7
(kW)
143
32
(kW)
13
2
(kW)
4.8
3
Pd
endloss
(kW)
23.5
2
endloss
(kW)
9
7
(cm 3)
2.5 x 106
1 x 108
p
P
trap
cx
Pbeam
cx
P
gas
nneutral (0)
36
.0277
-31-
ACKNOWLEDGEMENTS
I have received a great amount of assistance from several people in the
course of producing this work.
Dr.
Conrad at Wisconsin introduced me to the
Wolf code and helped with the initial neutral beam source design while Carol
Tull at the MFECC assisted in converting the code to run on the CRAY.
Drs.
Coleman and Torti at MIT, scientists in the TARA neutral beam group provided
much guidance throughout
the WOLF study.
Dr.
Futch at LLNL gave me the
buildup code and was always helpful in answering my questions.
I often
consulted with Dr. Kesner, TARA theory leader at MIT and compared my results
with those obtained by Dr.
acknowledge the support of Dr.
Horne at Wisconsin.
Post,
I would also like to
the leader of the TARA project who
suggested that I undertake this study.
Finally this report would not have
been possible without the help provided by Beth,
Elaine and Dorothea who
typed this paper and put up with my constant revisions (the curse of word
processing).
LIST OF FIGURES
Fig. 1
Electrode positions and ion beam trajectories for 20 keV hydrogen.
Due to symmetry only half the source is shown. RMS beam divergence is .0236
radians.
Fig. 2
Emitting surface for both 20 keV hydrogen and 25 keV deuterium for
an optimized (square points) and nearly optimized (diamond points) cases.
The surface is inputed and WOLF deforms the surface so as to obtain an
electric field at the surface of 300 V/cm.
Fig. 3
Electrode positions and ion beam trajectories for 25 keV
deuterium. RMS beam divergence is .0277 radians.
Fig. 4
Shows the sensitivity to beam divergence with electron temperature
for the two emitter surfaces shown in Fig. 2.
Fig. 5
Shows the RMS beam divergence with ion temperature for the
optimized emitter surface shown in Fig. 2. TARA source should have Ti from
1-2 eV.
Fig. 6
Shows RMS beam divergence for various current densities.
current density is .390 A/cm 2 .
Optimal
Fig. 7
Corresponding RMS electric field deviation from the desired value
of 300 V/cm.
Fig. 8
Shows RMS beam divergence as the distance between the emitter
surface and the acel electrode is varied.
Changes at points A, B and C are
caused by changing emitting surface input to achieve a numerical solution.
Fig. 9
Coarse mesh and fine mesh used to calculate beam divergences.
Fig. 10
Ion beam trajectories for 20 keV hydrogen using the coarse and
fine meshes.
Fig. 11
RMS beam divergence versus current density for coarse mesh (75
beamlets) 2and fine mesh (375 beamlets).
Optimal current density remains at
.390 A/cm , independent of mesh.
Fig. 12
Assumed time dependent behavior of the edge neutral pressure for
both hydrogen and deuterium base cases (Eq. 11).
Fig. 13
Initial ion density profiles for hydrogen and deuterium.
Fig. 14
Initial ion energy profiles for hydrogen and deuterium.
Fig. 15
Time evolution of both the ion density and energy profiles for the
base hydrogen case (20 keV hydrogen beams).
Fig. 16
Time evolution of both the ion density and energy profiles for the
base deuterium case (25 keV deuterium beams).
Fig. 17
cases.
Time evolution of the Franck-Condon neutral profiles for both base
Figures 18-60 are at t = 4 ms with the ratio of full
energy components being 1:2/3:1.
to half to third
Fig. 18
Hydrogen ion density profiles for various neutral beam positions.
Fig. 19
Hydrogen ion energy profiles for various neutral beam positions.
Fig. 20
Hydrogen Franck-Condon neutral profiles for various neutral beam
positions.
Fig. 21
Average electron density versus beam positioning for 20 keV
hydrogen beams.
Fig. 22
Total trapping power for 20 keV hydrogen beams (1.25 MW, 100A) for
various beam positions. For d - 10.5 cm, beam is close to plasma edge.
Fig. 23
Charge exchange with cold neutral gas power loss for 20 keV
hydrogen beams (1.25 MW, 100A) for various beam positions.
Fig. 24
Average electron density versus beam positioning for 25 keV
deuterium beams.
Fig. 25
Deuterium ion density profiles for various neutral beam positions.
Fig. 26
Deuterium ion energy profiles for various neutral beam positions.
Fig. 27
Deuterium Franck-Condon neutral profiles for various neutral beam
positions.
Fig. 28
Power loss through charge exchange with cold neutral gas for
various deuterium neutral beam positions.
Fig. 29
Total trapping power for 25 keV deuterium beams
for various beam positions.
(1.56
MW,
100 A)
Fig. 30
Hydrogen ion density profiles for various neutral beam currents
(50 A, 75 A, 100 A, 125 A and 150 A).
Fig. 31
currents.
Average electron density for various 20 keV hydrogen
beam
Fig. 32
Total beam trapping power for various 20 keV hydrogen
currents (0.62 MW, 0.94 MW, 1.25 MW, 1.56 MW and 1.87 MW).
beam
Fig. 33
Franck-Condon hydrogen neutral profiles for various
hydrogen beam currents.
Fig. 34
Power lost through charge exchange with cold gas
keV hydrogen beam currents.
Fig. 35
Global hydrogen ion particle confinement
hydrogen beam currents.
Fig. 36
20
keV
for various 20
time for various
20 keV
Deuterium ion density profiles for various neutral beam currents
(50 A, 75 A, 100 A, 125 A and 150 A).
Fig. 37
currents.
Average electron density for various 25 keV deuterium beam
Fig. 38
Franck-Condon deuterium neutral profiles for various 25 keV
deuterium beam currents.
Fig. 39
Power lost through charge exchange with cold gas for various 25
keV deuterium beam currents.
Fig. 40
Total beam trapping power for various 25 keV deuterium beam
currents (0,78 MW, 1.17 MW, 1.56 MW, 1.95 MW and 2.34 MW).
Fig. 41
Hydrogen ion density profiles for 100 A hydrogen neutral beam with
various energies (15 keV, 20 key, 25 keV and 30 keV).
Fig. 42
Average hydrogen ion density for various neutral beam energies.
Fig. 43
Hydrogen ion energy profiels for 100 A hydrogen neutral beam with
various energies.
Fig. 44
Hydrogen Franck-Condon neutral profiles for 100 A hydrogen neutral
beam with various energies.
Fig. 45
Deuterium ion density profiles for 100 A deuterium neutral beam
with various energies (15 keV, 20 keV, 25 keV and 30 keV).
Fig. 46
Average deuterium ion density for various neutral beam energies.
Fig. 47
Deuterium ion energy profiles
with various energies.
for 100 A deuterium neutral beam
Fig. 48
Deuterium Franck-Condon neutral profiles for 100 A deuterium
neutral beam with various energies.
Fig. 49
Total beam trapping power for 100 A deuterium neutral beams with
various energies (1.00 MW, 1.32 MW, 1.56 MW and 2.00 MW).
Fig. 50
Hydrogen ion density profile for 100
beam with various edge pressures from 3.0 x 10
, 20 keV hydrogen 8 neutral
Torr to 6.0 x 10.
Fig. 51
Hydrogen Franck-Condon neutral profile for 100 A,
neutral beam with various edge pressures.
20 keV hydrogen
Fig. 52
Hydrogen ion energy profile for 100 A,
beam with various edge pressures.
20 keV hydrogen neutral
Fig. 53
Deuterium ion density profile for 100 A
beeam with various edge pressures from 1.0 x 10
25 keV deuterium
Torr to 6.0 x 10
Fig. 54
Average electron density for 100 A,
with various edge pressures.
eutral
25 keV deuterium neutral beam
Fig. 55
Deuterium Franck-Condon neutral profile
deuterium neutral beam with various edge pressures.
for
100 A,
20 keV
Fig. 56
Hydrogen ion density profile for 100 A, 20 keV hydrogen beam with
the electron energy fixed at 100 eV, 200 eV, 500 eV, 750 kV and 1000 eV.
Fig. 57
Average electron density for 100 A, 20 keV hydrogen beam with the
electron energy fixed at 100 eV, 200 eV, 500 eV, 750 eV and 1000 eV.
Fig. 58
Hydrogen Franck-Condon neutral profiles for 100 A, 20 keV hydrogen
beam with the electron energy fixed at 100 eV, 200 eV, 500 eV, 750 eV and
1000 eV.
Fig. 59
Total beam trapping power for 100 A, 20 keV hydrogen beam with the
electron energy fixed at 100 eV, 200 eV, 750 eV and 1000 eV.
Fig. 60
Deuterium ion density profile for 100 A, 25 keV deuterium beam
with the electron energy fixed at 100 eV, 200 eV, 500 eV, 750 eV, 1000 eV.
Fig. 61
Average electron density for 100 A, 25 keV deuterium beam with the
electron energy fixed at 100 eV, 200 eV, 500 eV, 750 eV and 1000 eV.
Fig. 62
Deuterium Franck-Condon neutral profiles for 100 A, 25 keV
deuterium beam with the electron energy fixed at 100 eV, 200 eV, 500 ev, 750
eV and 1000 eV.
Fig. 63
Total beam trapping power for 100 A, 25 keV deuterium beam with
the electron energy fixed at 100 eV, 200 eV, 500 eV, 750 eV and 1000 eV.
Figures for Appendix with hydrogen (solid) and deuterium (dotted) curves.
Fig. A-1
Charge exchange cross section.
Fig. A-2
Ion impact ionization cross section.
Total beam trapping av (sum of charge exchange and ion and
Fig. A-3
electron impact ionization ov).
Fig. A-4
Charge exchange av.
Fig. A-5
Ion impact ionization ov.
Fig. A-6
Electron impact ionization av versus electron energy.
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